Lesson 1: Real Numbers & The Coordinate Plane
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Identify and classify real numbers (natural, whole, integers, rational, irrational)
- Understand the real number system hierarchy
- Plot points on the Cartesian coordinate plane
- Calculate distance between two points using the distance formula
- Find the midpoint between two points using the midpoint formula
The Real Number System
Before we dive into functions and graphs, we need to understand the foundation: real numbers. You've been working with numbers your whole life, but let's organize them into a clear hierarchy.
Real Numbers (ℝ) are all the numbers you can place on a number line. This includes positive numbers, negative numbers, zero, fractions, and even numbers that can't be written as fractions.
The Number Hierarchy
Real numbers are organized into categories, each one nested inside the next like Russian dolls:
All numbers on the number line
Can be written as a fraction a/b
..., -2, -1, 0, 1, 2, ...
0, 1, 2, 3, ...
1, 2, 3, 4, ...
Let's Define Each Category
Natural Numbers (ℕ)
Definition: The counting numbers
Set notation: {1, 2, 3, 4, ...}
Examples: 1, 5, 42, 1000
Whole Numbers (𝕎)
Definition: Natural numbers plus zero
Set notation: {0, 1, 2, 3, ...}
Examples: 0, 7, 15, 99
Integers (ℤ)
Definition: Whole numbers and their negatives
Set notation: {..., -2, -1, 0, 1, 2, ...}
Examples: -5, 0, 3, -100
Rational Numbers (ℚ)
Definition: Numbers that can be written as a/b where a and b are integers and b ≠ 0
Includes: All integers, fractions, terminating decimals, repeating decimals
Examples: 1/2, -3/4, 0.5, 0.333..., 7 (= 7/1)
Irrational Numbers
Definition: Numbers that CANNOT be written as a fraction
Characteristics: Decimals that never end and never repeat
Examples: √2, π (pi), e, √7, -√3
Real Numbers (ℝ)
Definition: All rational + all irrational numbers
Visual: Every point on the number line
Basically: every number you've ever encountered!
Example: Classifying Numbers
Let's classify each number and identify ALL categories it belongs to:
5
Natural, Whole, Integer, Rational, Real
0
Whole, Integer, Rational, Real (NOT Natural)
-3
Integer, Rational, Real (NOT Natural or Whole)
2/3
Rational, Real (can be written as a fraction!)
√7
Irrational, Real (approximately 2.645751..., never ends, never repeats)
π (pi)
Irrational, Real (3.14159..., never ends, never repeats)
The Cartesian Coordinate Plane
Now that we understand numbers, let's learn how to visualize relationships between them. The Cartesian coordinate plane (named after mathematician René Descartes) is a two-dimensional grid that lets us plot points and graph functions.
Cartesian Coordinate Plane: A two-dimensional plane formed by two perpendicular number lines (axes) that intersect at the origin.
Key Components
Origin (0, 0)
The point where the two axes intersect. This is the "home base" of the coordinate plane.
x-axis (horizontal)
The horizontal number line. Positive values go right, negative values go left.
y-axis (vertical)
The vertical number line. Positive values go up, negative values go down.
Ordered Pairs (x, y)
Coordinates that specify a point's location. The first number is x (horizontal), the second is y (vertical).
The Four Quadrants
The coordinate plane is divided into four regions called quadrants, numbered with Roman numerals:
Quadrant I
Location: Upper right
Signs: (+, +)
Example: (3, 5)
Quadrant II
Location: Upper left
Signs: (−, +)
Example: (-2, 4)
Quadrant III
Location: Lower left
Signs: (−, −)
Example: (-4, -3)
Quadrant IV
Location: Lower right
Signs: (+, −)
Example: (5, -2)
Memory Trick
Quadrants are numbered counterclockwise starting from the upper right (where both x and y are positive).
Distance Formula
How do we find the distance between two points on the coordinate plane? We use the distance formula, which comes from the Pythagorean theorem!
Distance Formula:
The distance d between points (x₁, y₁) and (x₂, y₂) is:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Example: Finding Distance
Find the distance between points (2, 3) and (5, 7).
Solution:
Step 1: Identify the coordinates
(x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 7)
Step 2: Substitute into the formula
d = √[(5 − 2)² + (7 − 3)²]
Step 3: Simplify
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5
Answer: The distance is 5 units.
Example: Another Distance Problem
Find the distance between points (−2, 3) and (4, −5).
Solution:
d = √[(4 − (−2))² + (−5 − 3)²]
d = √[(4 + 2)² + (−8)²]
d = √[6² + (−8)²]
d = √[36 + 64]
d = √100
d = 10
Answer: The distance is 10 units.
Midpoint Formula
The midpoint is the exact center point between two points. It's like finding the "halfway mark" on a line segment.
Midpoint Formula:
The midpoint M between points (x₁, y₁) and (x₂, y₂) is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In other words: average the x-coordinates, average the y-coordinates.
Example: Finding Midpoint
Find the midpoint between points (2, 3) and (8, 7).
Solution:
Step 1: Identify the coordinates
(x₁, y₁) = (2, 3) and (x₂, y₂) = (8, 7)
Step 2: Average the x-coordinates
x-coordinate of M = (2 + 8)/2 = 10/2 = 5
Step 3: Average the y-coordinates
y-coordinate of M = (3 + 7)/2 = 10/2 = 5
Answer: The midpoint is (5, 5).
Example: Midpoint with Negatives
Find the midpoint between points (1, 7) and (9, −3).
Solution:
M = ((1 + 9)/2, (7 + (−3))/2)
M = (10/2, 4/2)
M = (5, 2)
Answer: The midpoint is (5, 2).
Check Your Understanding
Try these questions to see if you've grasped the key concepts:
1. Classify the number 0. Which categories does it belong to?
2. Is √9 a rational or irrational number? Why?
3. In which quadrant is the point (−4, 7) located?
4. Find the distance between points (0, 0) and (3, 4).
d = √[(3 − 0)² + (4 − 0)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5 units
5. Find the midpoint between points (−2, 5) and (6, −1).
M = ((−2 + 6)/2, (5 + (−1))/2)
M = (4/2, 4/2)
M = (2, 2)
Key Takeaways
- Real numbers are organized into a hierarchy: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real
- Irrational numbers cannot be written as fractions and have decimals that never end or repeat (like √2, π)
- The Cartesian coordinate plane has two axes (x and y) that intersect at the origin (0, 0)
- The four quadrants are numbered counterclockwise: I (+,+), II (−,+), III (−,−), IV (+,−)
- Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
- Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) — average the coordinates!
Ready for More?
Next Lesson
In Lesson 2, you'll learn about relations and functions—one of the most important concepts in algebra. You'll discover what makes a function special and learn to use function notation!
Start Lesson 2Module Progress
You've completed Lesson 1! Keep going to build your algebra foundation.